Last updated at Feb. 4, 2020 by Teachoo

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Example 30 (Method 1) Find the coordinates of the point where the line through the points A (3, 4, 1) and B(5, 1, 6) crosses the XY-plane.The equation of a line passing through two points with position vectors ๐ โ & ๐ โ is ๐ โ = ๐ โ + ๐ (๐ โ โ ๐ โ) Given the line passes through the points (๐ โ โ ๐ โ) = (5๐ ฬ + 1๐ ฬ + 6๐ ฬ) โ (3๐ ฬ + 4๐ ฬ + 1๐ ฬ) = 2๐ ฬ โ 3๐ ฬ + 5๐ ฬ A (3, 4, 1) ๐ โ = 3๐ ฬ + 4๐ ฬ + ๐ ฬ B (5, 1, 6) ๐ โ = 5๐ ฬ + 1๐ ฬ + 6๐ ฬ โด ๐ โ = (3๐ ฬ + 4๐ ฬ + 1๐ ฬ) + ๐ (2๐ ฬ โ 3๐ ฬ + 5๐ ฬ) Let the coordinates of the point where the line crosses the XY plane be (x, y, 0). So, ๐ โ = x๐ ฬ + y๐ ฬ + 0๐ ฬ Since point crosses the plane, it will satisfy its equation Putting (2) in (1) x๐ ฬ + y๐ ฬ + 0๐ ฬ = 3๐ ฬ + 4๐ ฬ + 1๐ ฬ + 2๐๐ ฬ โ 3๐๐ ฬ + 5๐๐ ฬ x๐ ฬ + y๐ ฬ + 0๐ ฬ = (3 + 2๐)๐ ฬ + (4 โ 3๐)๐ ฬ + (1 + 5๐)๐ ฬ Two vectors are equal if their corresponding components are equal So, Solving 0 = 1 + 5๐ โด ๐ = (โ๐)/๐ So, x = 3 + ๐ = 3 + 2 ร (โ1)/5 = 3 โ 2/5 = 13/5 & y = 4 โ 3๐ = 4 โ 3 ร (โ1)/5 = 4 + 3/5 = 23/5 Therefore, the required coordinates are (๐๐/๐,๐๐/๐,๐) Example 30 (Method 2) Find the coordinates of the point where the line through the points A (3, 4, 1) and B(5, 1, 6) crosses the XY-plane.The equation of a line passing through two points A(๐ฅ_1, ๐ฆ_1, ๐ง_1) and B(๐ฅ_2, ๐ฆ_2, ๐ง_2) is (๐ โ ๐_๐)/(๐_๐ โ ๐_๐ ) = (๐ โ ๐_๐)/(๐_๐ โ ๐_๐ ) = (๐ โ ๐_๐)/(๐_๐ โ ๐_๐ ) Given the line passes through the points So, the equation of line is (๐ฅ โ 3)/(5 โ 3) = (๐ฆ โ 4)/(1 โ 4) = (๐ง โ 1)/(6 โ 1) A (3, 4, 1) โด ๐ฅ_1= 3, ๐ฆ_1= 4, ๐ง_1= 1 B(5, 1, 6) โด ๐ฅ_2 = 5, ๐ฆ_2= 1, ๐ง_2= 6 (๐ฅ โ 3)/2 = (๐ฆ โ 4)/(โ3) = (๐ง โ 1)/5 = k So, Since, the line crosses the XY plane at (x, y, 0) z = 0 5k + 1 = 0 5k = โ1 โด k = (โ๐)/๐ So, x = 2k + 3 = 2 ร (โ1)/5 + 3 = 3 โ 2/5 = 13/5 y = โ3 ร (โ1)/5 + 4 = 4 + 3/5 = 23/5 Therefore, the coordinates of the required point are (๐๐/๐, ๐๐/๐, ๐).

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Example 30 Important You are here

Chapter 11 Class 12 Three Dimensional Geometry (Term 2)

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.